Working
papers of Mark Joshi

We extend the limit optimal partial proxy method to compute second order sensitivities of financial products with discontinuous or angular payoffs by Monte Carlo simulation. The methodology is optimal in terms of minimizing the variance of likelihood ratios terms. Applications are presented for both equity options and interest rate products with discontinuous payoff structures. The first order optimal partial proxy method is also implemented to calculate the Hessians of insurance products with angular payoffs. Numerical results are presented which demonstrate the speed and efficacy of the method.

Variations of the binomial tree model are reviewed and extensions to the two most efficient trees studied in a recent literature are proposed. Tian’s modified tree is extended to a more general class of tree, and the third order tree is extended to the seventh order tree. Analysis of the error of American put pricing in the binomial tree model is presented and new trees that have superior performance in pricing in-the-money American puts are developed. To further improve numerical results, a scheme that incorporates different trees is suggested.

We derive the first known multiplicative dual for Bermudan type options that can be exercised more than once. Our multiplicative dual possesses the almost sure property, thus making it a viable alternative to the additive one. A method to compute the multiplicative upper bound is presented and the primal-dual algorithm is naturally extended to the multiplicative multiple-exercise case.

We analyze the primal-dual upper bound method and prove that its bias is inversely proportional to the number of paths in sub-simulations for a large class of cases. We develop a methodology for estimating and reducing the bias. We present numerical results showing that the new technique is indeed effective.

We discuss the pricing of cancellable swaps using the displaced diffusion LIBOR market model using a multi-core graphics card. We demonstrate that over one hundred times speed up can be achieved in a realistic case.

We present a new non-nested approach to computing additive upper bounds for callable derivatives using Monte Carlo simulation. It relies on the regression of Greeks computed using adjoint methods. We also show that it is is possible to early terminate paths once points of optimal exercise have been reached. A natural control variate for the multiplicative upper bound is introduced which renders it competitive to the additive one. In addition, a new bi-iterative family of upper bounds is introduced which take a stopping time, an upper bound, and a martingale as inputs.

We introduce a new approach to computing sensitivities of discontinuous integrals. The methodology is generic in that it only requires knowledge of the simulation scheme and the location of the integrand's singularities. The methodology is proven to be optimal in terms of minimizing the variance of the measure changes caused by the elimination of the discontinuities for finite bump sizes. An efficient adjoint implementation of the small bump-size limit is discussed, and the method is shown to be effective for a number of natural examples involving triggerable interest rate derivative securities.

We incorporate a simple and effective control-variate into Fourier inversion formulas for vanilla option prices. The control-variate used in this paper is the Black-Scholes formula whose volatility parameter is determined in a generic non-arbitrary fashion. We analyze contour dependence both in terms of value and speed of convergence. We use Gaussian quadrature rules to invert Fourier integrals, and numerical results suggest that performing the contour integration along the real axis leads to the best pricing performance.

In the framework of the displaced-diffusion LIBOR market model, we derive the pathwise adjoint method for the iterative predictor-corrector and Glasserman-Zhao drift approximations in the spot measure. This allows us to compute fast deltas and vegas under these schemes. We compare the discretisation bias obtained when computing Greeks with these methods to those obtained under the log-Euler and predictor-corrector approximations by performing tests with interest rate caplets and cancellable receiver swaps. The two predictor-corrector type methods were the most accurate by far. In particular, we found the iterative predictor-corrector method to be more accurate and slightly faster than the predictor-corrector method, the Glasserman-Zhao method to be relatively fast but highly inconsistent, and the log-Euler method to be reasonably accurate but only at low volatilities. Standard errors were not significantly different across all four discretisations.

We demonstrate how to compute first- and second-order sensitivities of portfolio credit derivatives such as synthetic collateralized debt obligation (CDO) tranches using algorithmic Hessian methods developed in Joshi and Yang (2010) in a single-factor Gaussian copula model. Our method is correct up to floating point error and extremely fast. Numerical result shows that, for an equity tranche of a synthetic CDO with 125 names, we are able to compute the whole Gamma matrix with computational times measured in seconds.

In this paper, we present an efficient approach to compute the first and the second order price sensitivities in the Heston model using the algorithmic differentiation approach. Issues related to the applicability of the pathwise method are discussed in this paper as most existing numerical schemes are not Lipschitz in model inputs. Depending on the model inputs and the discretization step size, our numerical tests show that the sample means of price sensitivities obtained using the Lognormal scheme and the Quadratic-Exponential scheme can be highly skewed and have fat-tailed distribution while price sensitivities obtained using the Integrated Double Gamma scheme and the Double Gamma scheme remain stable.

We discuss the issues involved in an efficient computation of the price and sensitivities of Bermudan exotic interest rate derivatives in the cross-currency displaced diffusion LIBOR market model. Improvements recently developed for an efficient implementation of the displaced diffusion LIBOR market model are extended to the cross-currency setting, including the adjoint-improved pathwise method for computing sensitivities and techniques used to handle Bermudan optionality. To demonstrate the application of this work, we provide extensive numerical results on two commonly traded cross-currency exotic interest rate derivatives: cross-currency swaps (CCS) and power reverse dual currency (PRDC) swaps.

We introduce a new methodology for computing Hessians from algorithms for function evaluation, using backwards methods. We show that the complexity of the Hessian calculation is a linear function of the number of state variables times the complexity of the original algorithm. We apply our results to computing the Gamma matrix of multi-dimensional financial derivatives including Asian Baskets and cancellable swaps. In particular, our algorithm for computing Gammas of Bermudan cancellable swaps is order O(n^2) per step in the number of rates. We present numerical results demonstrating that the computing all n(n+1)/2 Gammas in the LMM takes roughly n/3 times as long as computing the price.

This paper demonstrates how the adjoint PDE method can be used to compute Greeks in Markov-functional models. This is an accurate and efficient way to compute Greeks, where most of the model sensitivities can be computed in approximately the same time as a single sensitivity using finite difference. We demonstrate the speed and accuracy of the method using a Markov-functional interest rate model, also demonstrating how the model Greeks can be converted into market Greeks.

In this paper, we present three new discretization schemes for the Heston stochastic volatility model - two schemes for simulating the variance process and one scheme for simulating the integrated variance process conditional on the initial and the end-point of the variance process. Instead of using a short time-stepping approach to simulate the variance process and its integral, these new schemes evolve the Heston process accurately over long steps without the need to sample the intervening values. Hence, prices of financial derivatives can be evaluated rapidly using our new approaches.

We introduce two new methods to calculate bounds for zero-sum game options using Monte Carlo simulation. These extend and generalise the duality results of Haugh--Kogan/Rogers and Jamshidian to the case where both parties of a contract have Bermudan optionality. It is shown that the Andersen--Broadie method can still be used as a generic way to obtain bounds in the extended framework, and we apply the new results to the pricing of convertible bonds by simulation.

The problem of developing sensitivities of exotic interest rates
derivatives to the observed implied volatilities of caps and swaptions
is considered. It is shown how to compute these from sensitivities to
model volatilities in the displaced diffusion LIBOR market model. The
example of a cancellable inverse floater is considered.

We present a new method for truncating binomial trees based on using a
tolerance to control truncation errors and apply it to the Tian tree
together with acceleration techniques of smoothing and Richardson
extrapolation. For both the current (based on standard deviations) and
the new (based on tolerance) truncation methods, we test different
truncation criteria, levels and replacement values to obtain the best
combination for each required level of accuracy. We also provide
numerical results demonstrating that the new method can be 50% faster
than previously presented methods when pricing American put options in
the Black-Scholes model.

We introduce a new class of numerical schemes for discretizing
processes driven by Brownian motions. These allow the rapid computation
of sensitivities of discontinuous integrals using pathwise methods even
when the underlying densities post-discretization are singular. The two
new methods presented in this paper allow Greeks for financial products
with trigger features to be computed in the LIBOR market model with
similar speed to that obtained by using the adjoint method for
continuous pay-offs. The methods are generic with the main constraint
being that the discontinuities at each step must be determined by a
one-dimensional function: the proxy constraint. They are also generic
with the sole interaction between the integrand and the scheme being
the specification of this constraint.

We present a fast method to price and hedge CMS spread options in the
displaced-diffusion co-initial swap market model. Numerical tests
demonstrate that we are able to obtain sufficiently accurate prices and
Greeks with computational times measured in milliseconds. Further, we
find that CMS spread options are weakly dependent on the at-the-money
Black implied volatility skews.

Sensitivity analysis, or so-called 'stress-testing', has long been part
of the actuarial contribution to pricing, reserving and management of
capital levels in both life and non-life assurance. Recent developments
in the area of derivatives pricing have seen the application of adjoint
methods to the calculation of option price sensitivities including the
well-known 'Greeks' or partial derivatives of option prices with
respect to model parameters. These methods have been the foundation for
efficient and simple calculations of a vast number of sensitivities to
model parameters in financial mathematics. This methodology has yet to
be applied to actuarial problems in insurance or in pensions. In this
paper we consider a model for a defined benefit pension scheme and use
adjoint methods to illustrate the sensitivity of fund valuation results
to key inputs such as mortality rates, interest rates and levels of
salary rate inflation. The method of adjoints is illustrated in the
paper and numerical results are presented. Efficient calculation of the
sensitivity of key valuation results to model inputs is useful
information for practising actuaries as it provides guidance as to the
relative ultimate importance of various judgments made in the formation
of a liability valuation basis.

We introduce a new arbitrage-free interpolation scheme for the
displaced-diffusion LIBOR market model. Using this new extension, and
the Piterbarg interpolation scheme, we study the simulation of range
accrual coupons when valuing callable range accruals in the
displaced-diffusion LIBOR market model. We introduce a number of new
improvements that lead to significant efficiency improvements, and
explain how to apply the adjoint-improved pathwise method to calculate
deltas and vegas under the new improvements, which was not previously
possible for callable range accruals. One new improvement is based on
using a Brownian-bridge-type approach to simulating the range accrual
coupons. We consider a variety of examples, including when the
reference rate is a LIBOR rate, when it is a spread between swap rates,
and when the multiplier for the range accrual coupon is stochastic.

This paper derives
the pathwise adjoint method for the predictor-corrector drift
approximation in the displaced-diffusion LIBOR market model. We present
a comparison of the Greeks between log-Euler and predictor-corrector,
showing both methods have the same computational order but the latter
to be much more accurate.

We develop new
Monte Carlo techniques based on stratifying the stock's hitting-times
to the barrier for the pricing and Delta calculations of
discretely-monitored barrier options using the Black-Scholes model. We
include a new algorithm for sampling an Inverse Gaussian random
variable such that the sampling is restricted to a subset of the sample
space. We compare our new methods to existing Monte Carlo methods and
find that they can substantially improve convergence speeds.

We first develop an
efficient algorithm to compute Deltas of interest rate derivatives for
a number of standard market models. The computational complexity of the
algorithms is shown to be proportional to the number of rates times the
number of factors per step. We then show how to extend the method to
efficiently compute Vegas in those market models.

In this paper, we
present a generic framework known as the minimal partial proxy
simulation scheme. This framework allows stable computation of the
Monte-Carlo Greeks for financial products with trigger features via
finite difference approximation. The minimal partial proxy simulation
scheme can be considered as a special case of the partial proxy
simulation scheme (Fries and Joshi, 2008b) as a measure change
(weighted Monte Carlo) is performed to prevent path-wise
discontinuities. However, our approach differs in term of how the
measure change is performed. Specifically, we select the measure change
optimally such that it minimises the variance of the Monte-Carlo
weight. Our method can be applied to popular classes of trigger
products including digital caplets, autocaps and target redemption
notes. While the Monte-Carlo Greeks obtained using the partial proxy
simulation scheme can blow up in certain cases, these Monte-Carlo
Greeks remain stable under the minimal partial proxy simulation scheme.
Standard errors for Vega are also significantly lower under the minimal
partial proxy simulation scheme.

This paper extends
the pathwise adjoint method for Greeks to the displaced-diffusion LIBOR
market model and also presents a simple way to improve the speed of the
method. The speed improvements of approximately 20% are achieved
without using any additional approximations to those of Giles and
Glasserman.

We develop an
efficient algorithm to implement the adjoint method that computes
sensitivities of an interest rate derivative (IRD) with respect to
different underlying rates in the co-terminal swap-rate market model.
The order of computation per step of the new method is shown to be
proportional to the number of rates times the number of factors, which
is the same as the order in the LIBOR market model.

The calculation of
prices and sensitivities of exotic interest rate derivatives in the
LIBOR market model is often very time consuming. One approach that has
been previously suggested is to use a Markov-functional model as a
control variate for prices and deltas but not vegas. We present a new
approach that is effective for prices, deltas and vegas. It achieves a
standard error reduction by a factor of 10 for the price of a
five-factor, twenty-year Bermudan swaption, and of 5 for its vega.

We introduce a set
of improvements which allow the calculation of very tight lower bounds
for Bermudan derivatives using Monte Carlo simulation. These lower
bounds can be computed quickly, and with minimal hand-crafting. Our
focus is on accelerating policy iteration to the point where it can be
used in similar computation times to the basic least-squares approach,
but in doing so introduce a number of improvements which can be applied
to both the least-squares approach and the calculation of upper bounds
using the Andersen-Broadie method. The enhancements to the
least-squares method improve both accuracy and efficiency. Results are
provided for the displaced-diffusion LIBOR market model, demonstrating
that our practical policy iteration algorithm can be used to obtain
tight lower bounds for cancellable CMS steepener, snowball and vanilla
swaps in similar times to the basic least-squares method.

We
investigate the pricing performance of eight trinomial trees and one
binomial tree, which was found to be most effective in an earlier
paper, under twenty different implementation methodologies for pricing
American put options. We conclude that the binomial tree, the Tian
third order moment matching tree with truncation, Richardson
extrapolation and smoothing performs better than the trinomial trees.

The
pricing of snowball notes in the full-factor LIBOR market model is
considered. The primary aspect of the problem considered is the early
exercise feature, and it is shown how to characterize a class of
sub-optimal points of exercise. By combining this characterization with
least-squares regression on a suitable set of basis functions and using
an extra trigger enhancement, it is shown that very tight lower bounds
can be obtained in cases where previous methods required the use of
sub-Monte Carlo simulations.

In
this paper we present a generic method for the Monte-Carlo pricing of
(generalized) auto-callable products (aka. trigger products), i.e.,
products for which the payout function features a discontinuity with a
(possibly) stochastic location (the trigger) and value (the payout).
The
Monte-Carlo pricing of the products with discontinuous payout is known
to come with a high Monte-Carlo error. The numerical calculation of
sensitivities (i.e., partial derivatives) of such prices by finite
differences gives very noisy results, since the Monte-Carlo
approximation (being a finite sum of discontinuous functions) is not
smooth. Additionally, the Monte-Carlo error of the finite-difference
approximation explodes as the shift size tends to zero. Our
method combines a product specific modification of the underlying
numerical scheme, which is to some extent similar to an importance
sampling and/or partial proxy simulation scheme and a reformulation of
the payoff function into an equivalent smooth payout. From
the financial product we merely require that hitting of the stochastic
trigger will result in an conditionally analytic value. Many complex
derivatives can be written in this form. A class of products where this
property is usually encountered are the so called auto-callables, where
a trigger hit results in cancellation of all future payments except for
one redemption payment, which can be valued analytically, conditionally
on the trigger hit. From the model we require that its numerical
implementation allows for a calculation of the transition probability
of survival (i.e., non-trigger hit). Many models allows this, e.g.,
Euler schemes of Itô processes, where the trigger is a model primitive.
The method presented is effective across a large range of cases where
other methods fail, e.g. small finite difference shift sizes or short
time to trigger reset (approaching maturity); this means that a
practitioner can use this method and be confident that it will work
consistently.

We introduce a new
calibration methodology that allows perfect fitting of the displaced
diffusion LIBOR market model to caplets and co-terminal
swaptions, whilst avoiding global optimizations. The approach works by
regarding a forward rate as a difference of swap-rates and then
bootstrapping through rates one by one.

Various drift approximations for the
displaced-diffusion LIBOR market model in the spot measure are
compared. The advantages, disadvantages and implementation choices for
each of predictor-corrector and the Glasserman--Zhao method are
discussed. Numerical tests are carried out and we conclude that the
predictor-corrector method is superior.

Abstract: We study 20
different implementation methodologies for each of 11 different choices
of parameters of binomial trees and investigate the speed of
convergence for pricing American put options numerically. We conclude
that the most effective methods involve using truncation, Richardson
extrapolation and sometimes smoothing. We do not recommend use of a
European option as a control. The most effective trees are the Tian
third order moment matching tree and a new tree designed to minimize
oscillations.

Abstract: A new family of
binomial trees as approximations to the Black--Scholes model
is introduced. For this class of trees, the existence of complete
asymptotic expansions for the prices of vanilla European options is
demonstrated and the first three terms are explicitly computed. As
special cases, a tree with third order convergence is constructed and
the conjecture of Leisen and Reimer that their tree has second order
convergence is proven.

Abstract: We
consider a generic framework which allows to calculate robust
Monte-Carlo sensitivities seamlessly through simple finite difference
approximation. The method proposed is a generalization and improvement
of the proxy simulation scheme method (Fries and Kampen, 2005). As a
benchmark we apply the method to the pricing of digital caplets and
target redemption notes using LIBOR and CMS indices under a LIBOR
Market Model. We calculate stable deltas, gammas and vegas by applying
direct finite difference to the proxy simulation scheme pricing. The
framework is generic in the sense that it is model and almost product
independent. The only product dependent part is the specification of
the proxy constraint. This allows for an elegant implementation, where
new products may be included at small additional costs

Abstract:
A new binomial approximation to the Black–Scholes model is
introduced. It is shown that for digital options and vanilla European
call and put options that a complete asymptotic expansion of the error
in powers of 1/n exists. This is the first binomial tree for which such
an asymptotic expansion has been shown to exist.

Abstract: We develop a
completely new model for correlation of credit defaults based on a
financially intuitive concept of business time similar to that in the
Variance Gamma model for stock price evolution. Solving a simple
equation calibrates each name to its credit spread curve and we show
that the overall model can be calibrated to the market base correlation
curve of a tranched CDO index. Once this calibration is performed,
obtaining consistent arbitrage-free prices for non-standard tranches,
products based on different underlying names and even more exotic
products such as $\mathrm{CDO}^2$ is straightforward and rapid.

Abstract.
The problem of pricing a
continuous barrier option in a jump-diffusion model is studied. It is
shown that via an effective combination of importance sampling and
analytic formulas that substantial speed ups can be achieved. These
techniques are shown to be particularly effective for computing deltas.

Abstract: An algorithm of
order
number of factors times number of rates for the computing the drifts of
all the rates in the LIBOR market model. This is better than the naive
algorithm which is of order number of rates squared.

Abstract.
The pricing of callable
derivative products with complicated pay-offs is studied. A new method
for finding upper bounds by Monte Carlo simulation is introduced, this
relies on modelling the callable product directly. The method has a
wide range of applicability and is shown to be effective for Asian tail
products. Presentation

Abstract. An algorithm
for computing the drift in the LIBOR market model with additional
idiosynchratic terms is introduced. This algorithm achieves a
computational complexity of order equal to the number of common factors
times the number of rates. It is demonstrated that this allows better
matching of correlation matrices in reduced-factor models.

Abstract. Rogers’
method for upper bounds for Bermudan options is rephrased in terms of
buyers and sellers prices. It is shown how to deduce
Jamshidian’s upper bound result in a simple fashion from
Roger’s method, including the case of possibly zero final
pay-off. Both methods are improved by ruling out exercise at suboptimal
points. It is also shown that it is possible to use sub- Monte Carlo
simulations to estimate the value of the hedging portfolio at
intermediate points in the Jamshidian method without jeopardizing its
status as upper bound.

Abstract. A number of standard
market models are studied. For each one, algorithms of computational
complexity equal to the number of rates times the number of factors to
carry out the computations for each step are introduced. Two new
classes of market models are developed and it is shown for them that
similar results hold.

Abstract: We present four new
methods for approximating the drift in the LIBOR market model. These
are compared to a variety of existing methods including PPR,
Glasserman-Zhao and predictor-corrector. We see that two of them which
use correlation adjustments to better approximate the drift are more
effective than existing methods.

It is well-known that the
Dirichlet problem for the Laplacian on a reasonably smooth compact
domain in Rn can be solved using Brownian motion. Indeed the result was
found by Kakutani in 1944, [3, 4]. In this note, I want to discuss how
this result can be reinterpreted financially. Our objective is to
increase our intuition about the problem rather than to attempt to
prove new results.

Advice

Here's my advice sheet for those wanting
to work as a quantitative analyst in finance. It was originally aimed
at pure maths PhDs, but lots of other people seem to like it.

Software downloads

I have now released xlw 2.1 formerly called xlwPlus. xlw is a package for creating xlls in C++ with
minimal
effort. An xll is a way of adding new functions to EXCELxlwPlus
has various added features, the most important is that the interfacing
code is generated automatically by an extra routine instead of being
coded manually. If you have any questions, please ask them on xlw-users
mailing list on sourceforge. Eric Ehlers has now updated xlw to contain
the new Excel Interface. Narinder Claire and John Adcock have provided further improvements which have increased user-friendliness.
The latest version is 5.1.

The code for C++ design
patterns and derivatives
pricing is available here.